Problem: Let $f(x)=3x+4$ and $g(x)=2x-3$.  If $h(x)=f(g(x))$, then what is the inverse of $h(x)$?
Solution: \[h(x)=f(g(x))=3(2x-3)+4=6x-5.\]Let's replace $h(x)$ with $y$ for simplicity, so \[y=6x-5.\]In order to invert $h(x)$ we may solve this equation for $x$.  That gives \[y+5=6x\]or  \[x=\frac{y+5}{6}.\]Writing this in terms of $x$ gives the inverse function of $h$ as  \[h^{-1}(x)=\boxed{\frac{x+5}{6}}.\]